Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the low-lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and L-functions.

By looking at the average behavior (n-level density) of the low lying zeros of certain families of L-functions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the non-trivial zeros in terms of the classical compact groups. This is further supported by numerical experiments for which an efficient algorithm to compute L-functions was developed and implemented. iii Acknowledgements When Mike Rubinstein woke up one morning he was shocked to discover that he was writing the acknowledgements to his thesis. After two screenplays, a 40000 word manifesto, and many fruitless attempts at making sushi, something resembling a detailed academic work has emerged for which he has people to thank. Peter Sarnak- from Chebyshev's Bias to USp(1). For being a terrific advisor and teacher. For choosing problems suited to my talents and involving me in this great project to understand the zeros of L-functions. Zeev Rudnick and Andrew Oldyzko for many disc...

Abstract. Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli k ≤ 72 and other small moduli. 1.

The Generalized Riemann Hypothesis (GRH) states that all non-trivial zeros of Dirichlet L-functions lie on the line Re(s) = 1 2. Further, it is believed that there are no Q-linear

Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multi-dimensional "hypergeometric method" for rational and algebraic approximation to algebraic numbers. Consequently, if a; b and n are given positive integers with n 3, we show that the equation of the title possesses at most one solution in positive integers x; y. Further results on Diophantine equations are also presented. The proofs are based upon explicit Pad'e approximations to systems of binomial functions, together with new Chebyshev-like estimates for primes in arithmetic progressions and a variety of computational techniques. 1.

Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree-1 prime p of K such that p = σ, satis-

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Abstract. In this paper, we describe a computation which established the GRH to height 92 (resp. 40) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff error to obtain results which are mathematically rigorous, and we generalize Turing’s criterion to prove that there is no zero off the critical line. We finally give results concerning the GRH for cubic and quartic fields, tables of low zeros for number fields of degree 5 and 6, and statistics about the smallest zero of a number field. 0. Introduction and notations The Riemann zeta function and its generalization to number fields, the Dedekind zeta function, have been for well over a hundred years one of the central tools in number theory. It is recognized that the deepest single open problem in mathematics is the settling of the Riemann Hypothesis, and number theorists know that its

A small part of the Generalized Riemann Hypothesis asserts that L-functions do not have zeros on the line segment ( 1

Abstract. This article improves the estimate of the size of the definite integral of S(t), the argument of the Riemann zeta-function. The primary application of this improvement is Turing’s Method for the Riemann zeta-function. Analogous improvements are given for the arguments of Dirichlet L-functions and of Dedekind zeta-functions. 1.